Non-equivalent Atomic Vibrations at Interfaces in a Polar Superlattice

In heterostructures made from polar materials, e.g. , AlN-GaN-AlN, the non-equivalence of the two interfaces has long been recognized as a critical aspect of their electronic properties, in that they host different two-dimensional carrier gasses. Interfaces play as important a role in the vibrational properties of materials, where interface states enhance thermal conductivity and can generate unique infrared-optical activity. The non-equivalence of the corresponding interface atomic vibrations, however, has not been investigated so far due to a lack of experimental techniques with both high spatial and high spectral resolution. Herein we experimentally demonstrate the non-equivalence of AlN-(Al 0.65 Ga 0.35 )N and (Al 0.65 Ga 0.35 )N-AlN interface vibrations using monochromated electron energy-loss spectroscopy in the scanning transmission electron microscope (STEM-EELS) and employ density-functional-theory (DFT) calculations to gain insights in the origins of observations. We demonstrate that STEM-EELS possesses sensitivity to the displacement vector of the vibrational modes as well as the frequency, which enables direct mapping of the non-equivalent interface phonons between materials with different phonon polarizations in the interface direction. The physical origin of the non-equivalent interface behavior is then unraveled by understanding the localization and anisotropy of phonon displacements using DFT calculations. The results demonstrate the capacity to carefully assess the vibrational properties of complex heterostructures where interface states dominate the functional properties.


Abstract
In heterostructures made from polar materials, e.g., AlN-GaN-AlN, the non-equivalence of the two interfaces has long been recognized as a critical aspect of their electronic properties, in that they host different two-dimensional carrier gasses.Interfaces play as important a role in the vibrational properties of materials, where interface states enhance thermal conductivity and can generate unique infrared-optical activity.The non-equivalence of the corresponding interface atomic vibrations, however, has not been investigated so far due to a lack of experimental techniques with both high spatial and high spectral resolution.Herein we experimentally demonstrate the non-equivalence of AlN-(Al0.65Ga0.35)N and (Al0.65Ga0.35)N-AlNinterface vibrations using monochromated electron energy-loss spectroscopy in the scanning transmission electron microscope (STEM-EELS) and employ density-functional-theory (DFT) calculations to gain insights in the origins of observations.We demonstrate that STEM-EELS possesses sensitivity to the displacement vector of the vibrational modes as well as the frequency, which enables direct mapping of the non-equivalent interface phonons between materials with different phonon polarizations in the interface direction.The physical origin of the non-equivalent interface behavior is then unraveled by understanding the localization and anisotropy of phonon displacements using DFT calculations.The results demonstrate the capacity to carefully assess the vibrational properties of complex heterostructures where interface states dominate the functional properties.

Introduction
][9][10][11][12] The emergent interface electronic and vibrational structures depend on the relative order of materials.For example, (Al1-xGax)N-AlN and AlN-(Al1-xGax)N interfaces feature different two-dimensional electron gasses, [13][14][15] and, similarly, non-equivalent interface atomic vibrations from charge-lattice coupling have been proposed. 12However, directly probing the non-equivalent interface atomic vibrations is challenging due to the limited availability of techniques with both the high-spatial resolution and high-energy resolution that are needed to identify an interface and its vibrational response.Knowledge about how, and if, chargelattice interactions lead to non-equivalent interface atomic vibrations would provide a deeper understanding of how interfaces impact phonon mediated properties.
7][18][19][20][21] This resolution has proven useful in understanding how local structure and chemistry impact atomic vibrations for a range of defects from zero-dimensional point defects 20,22 , to one-dimensional grain boundary dislocations 23 , and to two-dimensional boundaries [24][25][26][27] .Such spatially resolved measurements have proven powerful in understanding how structurally 8 or chemically 9,28 diffuse interfaces impact properties like thermal boundary conductance, which is dominated by localized interface vibrations.Wu et al. 29 have also demonstrated that the Al stoichiometry in a (Al1-xGax)N-AlN heterostructure determines the spatial extent of phonon amplitudes.However, since these works have only examined bilayer heterostructures, only single interfaces have been present in the samples and the non-equivalence of atomic vibrations at seemingly similar interfaces in heterostructures with polar materials has not been investigated.
Herein, we examine a prototypical AlN-(Al0.65Ga0.35)Nsuperlattice using aberration-corrected and monochromated STEM-EELS combined with density-functional-theory (DFT) calculations to distinguish the atom vibrations at AlN-(Al0.65Ga0.35)Nand (Al0.65Ga0.35)N-AlNinterfaces and understand how they differ.We observe that the AlN-(Al0.65Ga0.35)Ninterface exhibits phonon softening behavior while the (Al0.65Ga0.35)N-AlNinterface does not, which demonstrates that the two seemingly similar interfaces are vibrationally non-equivalent.Using the eigenvector displacement selectivity of off-axis EELS we also demonstrate that the vibrations of atoms at the interfaces are distinct at the two interfaces along the direction perpendicular to the interface but equivalent parallel to the interface.This directionally resolved displacements are related to the structure of the interfaces using DFT and demonstrate that the atom vibrations at the interfaces are not only distinct but also highly anisotropic in nature.

Results
Figure 1(a) shows a schematic of a DFT-calculated relaxed (AlN)2-(GaN)2 superlattice that contains two unit cells of GaN, two unit cells of AlN, and two interface planes.Distinct structural units can be defined at the two interface planes bounding the III-N layers, which are labeled as top and bottom interface in Figure 1(b).In Figure 1(a,b) the individual Al/Ga-N out-of-plane dipoles are highlighted with blue and green lines respectively.In the top-interface structural unit, the AlN dipole is at the top of the unit while in the bottom-interface structural unit it is at the bottom of the unit.This non-equivalence results in differences in the dipole bond length as shown in Figure 1(c), where the interface dipoles are distinguishable from the layers. 4In general, the structure results in a non-equivalence of the two interfaces.Here, we examine an AlN-(Al0.65Ga0.35)Nsuperlattice, grown using metalorganic chemical vapor deposition.Superlattices have the advantage of possessing many interfaces to both access the interface non-equivalence and to establish the reproducibility of the measurement.The HAADF image in Figure 1(d) shows an overview of the superlattice and its periodic nature.In Figure 1(e), a higher resolution image shows the quality of the top and bottom-interfaces of one (Al0.65Ga0.35)Nlayer.This layered III-V structure has the same structural phenomena occurring as in the pure AlN-GaN superlattices shown in Figure 1(a-c) but with different magnitudes, as shown in Figure S1.
We also note that structural variability is not observed in-plane (see Figure S2).
We then use high-spatial/spectral resolution EELS to understand how the unique interface atomic vibrations are impacted by non-equivalent interface structures.Figure 1(f) shows a schematic of the off-axis geometry that enables localized spatial mapping by scanning a converged electron probe over a sample and then filtering the scattered electrons.The scattering cross-section in an off-axis condition may be written as 24,28 where ω is frequency, j is a band index, κ is the atom index, and e are eigenvector displacements.
This cross-section demonstrates that the scattered electron probe can resolve individual modes at band j and q within the experimental uncertainty Δq of the converged probe . 30,31More critically, Equation 1 demonstrates, that the eigenvectors of individual atomic displacements in a phonon mode couple to the net momentum selected by the aperture.The full Equation 1 is found in section S4 and can also provide information about phonon transport. 28In a large convergence-angle condition (i.e.large Δq) off-axis EELS loses sensitivity to modes in a Brillouin zone, effectively being integrated into a density of states (DOS).However, the loss of selectivity to states with discrete crystal momenta does not mean that selectivity to states with net momentum larger than the convergence angle and parallel to an eigenvector displacement is lost.This momentum sensitivity to eigenvector displacements in off-axis EELS with a large convergence angle is, therefore, proportional to the projected phonon DOS (PPhDOS; see Methods and section S4).Therefore, the electron probe has some reciprocal-space selectivity to vibrational modes despite the lack of momentum resolution from the highly converged probe.This selectivity of modes allows for the analysis of modes where atomic vibrations exhibit a particular net momentum transfer and possible anisotropies.the top interface is consistent with data reported by Wu et al. 29 , which showed similar interface behavior in a bilayer sample.This swooping feature in the data can be thought of as the change dipole interatomic distances (Figure 1(c)) acting similar to a c-axis strain, which is known to have a large impact on the softening of phonon modes. 32e alloy and N vibrational modes are integrated over 8 meV windows and their direct intensity  To better understand the spatial symmetry of the vibrations we take energy-versus-intensity profiles at energies relating to pure Ga, III, Al, and N vibrations (Figure 2(g-j)).Furthermore, the width of all the profiles is like the ADF demonstrating that the measured response is localized to the probe and not resulting from delocalized aloof interactions.The low-energy Ga profile and high-energy N profile are symmetric for both scattering conditions.This feature suggests that neither the N nor Ga dominant phonons are significantly modified by the presence of the interfaces, nor is there any variation in the total signal that could account for the interface non-equivalence.
The same isotropic nature is observed for the in-plane scattering direction at III and Al vibration dominated energies, which reiterates the previous conclusion that the in-plane behavior reflects the isotropy of the superlattice and wurtzite crystal structure.For the qz conditions normal to the interfaces, we clearly see the spatial asymmetry of interface vibrations in the III and Al vibration profiles.The extrema in the profiles coincide with the interfaces in the superlattice period exhibiting a peak at the top interface and a dip at the bottom interface.
The DFT-calculated PPhDOS, demonstrates good agreement with the experimental line profile and provides a direct connection to the atomistic structure of the interfaces.By comparing the qx (Figure 2(e)) and qz (Figure 2(f))PPhDOS, we see that both exhibit some slight asymmetry between the top and the bottom interfaces, but the effect is far more pronounced in the qz PPhDOS.
The PPhDOS shows that the asymmetry observed at these interfaces stems from an intrinsic difference in the local vibrational density of states across the material that is detectable via the nature of the electron beam-vibration interaction.
To unravel the complex vibrational behavior that we are observing in projected momentum directions and locally in space we turn to DFT phonon calculations of the structure shown in Figure 1(a).The dispersion curves in Figure 3(a-c) show details pertaining to each phonon mode in a (11 ̅ 00) plane, which is the same projection as Figure 2(a).We quantify the general spatial localization of a phonon mode by calculating each mode's participation ratio (PR), as shown in Figure 3(a) and expanded upon in methods.A PR of 1 is a completely delocalized mode while a PR=1/N corresponds to maximum localization to one atom, where N is the number of atoms in the system, which here refers to one period of the superlattice.We see that many of the lower-energy modes are highly delocalized while some modes, particularly the momentum path L-H, are localized.The high-energy modes are generally more localized then the low-energy and some approach the 1/N limit.Overall, Figure 3(a) demonstrates that there is a high degree of localization to particular atoms within particular periodic layers in the phonon modes of the superlattice.To understand how this plot relates to EELS, we compare the PPR to the mean-square displacement along the z-axis (|uz| 2 ), as shown in Figure 3(c), to highlight which vibrational modes will have eigenvector displacements that can be sampled with the qz collection.Here we see that there are very large atomic displacements for many of the modes that are localized in Figure 3(a,b) and some that are not localized at the top-interface but have very large |uz| 2 .
The superlattice modes falling in or near the GaN gap show unique characteristics in Figure 3(ac) (also see Figure S9 and Section S3).We select four of these modes with variable degrees of localization or large |uz| 2 and show the displacement vectors on a model alongside a plot of the magnitudes in Figure 3(d-g).In all four, the z-component has a large displacement amplitude at the top-interface relative to the z-component of atomic displacements elsewhere.This large z-component does not necessarily mean only interface atoms are displacing in a mode, i.e., that the mode is localized, but rather signifies that the z-displacement at the top-interface is dominant relative to the other z-displacements.In Figure 3(d) the phonon has a very large displacement at the interface and decays into the AlN layer.In Figure 3(e,f) the mode is delocalized within the AlN layer and interfaces.However, the z-component is localized at the top-interface, being peaked at the topinterface for z and then progressing to x displacements in the layer.Therefore, preferentially selecting the z-component in an off-axis EELS condition gains sensitivity to the local vibration at that interface and to the interface non-equivalence.Cross-sectional STEM samples used in the 4D-STEM experiments were made with traditional mechanical polishing.The sample was first sandwiched together then a dimple was ground along the glue line.An Allied MultiPrep TM was used to polish the sample to 1 μm then a second-generation Gatan precision ion polishing system was used to thin the sample to electron transparency.
Cross-sectional STEM samples used in the monochromated STEM-EELS experiments were made using a Thermo Fisher Scientific Helios Dual Beam focused ion-beam.Initial milling and cleaning were performed at 30 kV, which was sequentially decreased until a finishing energy of 2 kV.

STEM Imaging and 4D-STEM
STEM imaging in the main text and 4D-STEM acquisitions shown in the supplementary information was acquired using a Nion UltraSTEM100.CoM 4D-STEM acquisitions shown in Figure S10 used a 30 mrad semi-convergence angle.Calculated CoM vector images are decomposed to a divergence-free and curl-free component using Helmholtz decomposition 33 for interpretation.The PACBED in Figure S11 were acquired with a 10 mrad semi-convergence angle.

Vibrational EELS
Vibrational EELS spectra were acquired at 100 keV using a Nion HERMES monochromated aberration-corrected dedicated STEM with a convergence angle of 30 mrad and ~32 mrad entrance aperture collection angle.Two off-axis conditions were selected by tilting the electron beam after the sample and before the spectrometer entrance aperture.
The zero-loss peak in each I(E) spectrum along the q-axis was then aligned.We found that crosscorrelation of the zero-loss peak lead to the most symmetric and best resolution zero-loss peak.
The resolution for the non-aligned versus aligned and brightfield versus darkfield are found for both off-axis conditions in Figure S12 and Figure S13.All alignment methods resulted in a zeroenergy spike due to far off-axis I(E) spectrum with single counts being aligned to zero.This did not affect the non-zero spectral response because the misaligned spectra only contained one count.
The alignment improves the spectral resolution defined by the full-width at half-maximum after masking the zero-loss spike followed by interpolating .
Spectra were acquired as four-dimensional datasets, with one temporal two spatial (x, y), one perpendicular momentum (q), and one energy-loss (E) dimension.The x-spatial dimension parallel to the interfaces was use instead of a time series to provide better counting and spatial statistics by aligning then averaging the x-axis resulting in the profiles shown in Figure 1(h) and Figure S4.
Projection of each line scan was then performed by 1) background subtracting thickness effects from the simultaneously acquired ADF signal, 2) peak finding, 3) windowing 12 nm around each peak, 4) cross-correlating each window then aligning for better period-to-period registration, and then 5) cropping to the 10 nm period thickness.The spatial correlation and cropping found from the ADF was then applied to the EELS line scans.

Calculations
All simulations were performed within the local density approximation (LDA) framework of density functional theory (DFT), utilizing the Vienna Ab-initio Simulation Package (VASP).The electronic configurations of gallium (Ga), nitrogen (N), and aluminum (Al) were described by projector augmented wave (PAW) pseudopotentials.
The calculational cell of (AlN)2-(GaN)2 superlattice contains 16 atoms, with relaxed lattice constants of a = 6.Phonon dispersion calculations were conducted via the finite-displacement method, employing Phonopy, 34 an open-source package.The superlattice phonons were calculated using a 3x3x2 supercell with 288 atoms.The cutoff energy was set at 800 eV.The energy-convergence threshold was set 10 -8 eV, using a 4x4x1 gamma-centered k-point mesh.For phonons of bulk AlN and GaN, 4x4x2 supercells, each containing 128 atoms, were modeled, using a 6x6x6 gamma-centered kpoint mesh, adhering to the same energy cutoff and convergence criteria.
To calculate the dipole PPhDOS, the element PPhDOS was first calculated then the III and N atoms in each dipole were summed.The PPhDOS were then blurred with a Gaussian to match the finite energy resolution of experiments.A series of spectral blurring is found in Figure S5 -Figure S7.
Subsequent analysis of the PPhDOS, PR, PPR, and displacement models were performed using a combination of Phonopy and custom python scripts.We define the participation ratio as

𝑁 𝑢
where Nu refers to the number of atoms in corresponding structural unit.
The atomic displacement from each mode can be calculated via where   = ( ℏ     ⁄ − 1) −1 being the Bose-Einstein distribution function, ℏ is the reduced Plank's constant, and T is temperature.

Figure 1 .
Figure 1.The non-equivalent structure of III-V heterostructure interfaces.(a) A ball-andstick model illustrating a single period of a (AlN)2-(GaN)2 superlattice.The superlattice period is indicated with the solid lines while dashed red lines mark the interfaces.(b) Structural units present in a single period of a superlattice, each of which are separated by grey lines and labeled on the right-hand side.(c) Bond length of the III-N dipole emphasizing the unique state of each structural unit.Red-dashed lines indicate the interface planes in (a), and black-dashed lines divide structural unit in (b).The average bond length for the AlN (blue) and GaN (green) layers provide a reference for the interface displacements.(d) A high-angle annular darkfield (HAADF) image of an AlN-(Al0.65Ga0.35)Nsuperlattice showing bright (Al0.65Ga0.35)Nlayers and dark AlN layers.The spatial coordinates referencing vectors normal to the SL (z) and parallel to the SL (x) are annotated.(e) Higher resolution HAADF image that focuses on a single period of the superlattice and emphasizes the interfaces.The top and bottom-interfaces of the (Al0.65Ga0.35)Nare labeled and annotated in cyan.(f) Schematic diagram of a STEM off-axis EELS experiment where the scattered probe is tilted away from the optic-axis (green dotted) to an off-axis condition such that the spectrometer entrance aperture collects scattering space qn, where n refers to x-and z-directions.(g,h) Off-axis STEM-EELS line profile.(g) ADF profile to provide context to where in the sample the off-axis spectra in (h) are being acquired from.Higher intensity regions in the ADF are (Al0.65Ga0.35)Nand

Figure 1 (
Figure 1(g,h)show the off-axis ADF and EELS profile acquired over approximately ten periods with the aperture collecting qz (i.e., only electrons scattered along the superlattice axis).Two primary bands at 35-45 meV and 80-95 and a fainter band at 60-70 meV are observed.Elementresolved, projected phonon density of states (PPhDOS) are performed to assign these peaks to vibrational modes (FigureS3) and show that the highest energy band is dominated by N vibrations, the middle band by Al vibrations, the lower band by the simultaneous vibration of both III site atoms, and everything less than 35 meV as dominated by the Ga vibrations.Note that III site vibrations do not necessarily refer to an alloy vibration but may also refer to Al and Ga vibrating in separate layers, hence why the band is observed in both layers.
profiles over two periods are shown in Figure 1(i).The N vibrations are symmetric about the (Al0.65Ga0.35)Nlayers while the III-atom vibrations are asymmetric about the (Al0.65Ga0.35)Nlayers, i.e., showing an inversion at the center of each (Al0.65Ga0.35)Nlayer, where the peaks and dips of the signature vibrational response of the interface non-equivalence are annotated with arrows.Therefore, the inherent relation between structure and atomic vibrations results in unique interface III-atom vibrations at the AlN/(Al0.65Ga0.35)Nand (Al0.65Ga0.35)N/AlNinterfaces.

Figure 2 .
Figure 2. Angularly and spatially resolved vibrational response of a polar heterostructure.Reference (a) PACBED pattern annotating two darkfield conditions for separate vibrational EELS spectrum images.Scattering conditions annotated with qz are normal to the interfaces and along the III-V dipole while qx are in-plane.Representative period-projected (b) ADF profile to provide context to where in the sample the spectra are being acquired, with higher intensity (I) being (Al0.65Ga0.35)Nand lower intensity AlN.Period-projected off-axis EELS line scans for the (c) qx and (d) qz scattering.The original scans are found in Figure S4.The EELS spectra intensities are normalized by multiplying by energy-loss (E) to remove the elastic background and power law scaled to bring out lower intensity features.Dipole PPhDOS along the (e) qx and (f) qz axes that has been spectrally blurred to the finite energy resolution of experiments.(g-j) Intensity profiles as a function of position across the projected period for specific energy slices.Intensities are normalized relative to AlN and scaled from ±1, i.e., zero being the intensity of vibrational response in the AlN layer.Grey dotted lines indicate the approximate position of the interfaces as determined by the full-width-half-max of the ADF signal (black dash-dot).

Figure 3 .
Figure 3. Unraveling vibrational dynamics at non-equivalent interfaces using DFT.(a) Phonon dispersion with the color indicating the PR.(b) Layer PPR with the colors indicating the layer's relative contribution to the total PR.Red (R) indicates localization within the layers, green (T) at the top-interrace, and blue (b) at the bottom-interface.(d-g) Shows the displacements on a model (left) and the xz-projected displacement magnitudes for the modes circled in (a-c).The models and profiles are centered on the AlN instead of the GaN to emphasize the amplitude profile in the AlN.The interfaces at the top and bottom of the GaN are annotated with black dotted lines.

Figure 3 ( 4 Conclusions
g) depicts the case of a mode that is both highly localized and has the displacements purely along the z-axis.Each of these four examples showcase the non-equivalent vibrational displacements along the z-axis at the interfaces that are inherently coupled to the structural modifications at the two interfaces shown in Figure1and are detectable via angular scattering as shown in Figure2.In a heterostructure containing polar materials, the interface characteristics greatly depend on the relative order of the bounding materials.Here, we show in III-N superlattices that the structure and atomic vibrations of AlN-(Al0.65Ga0.35)Nand (Al0.65Ga0.35)N-AlNinterfaces are not equivalent using off-axis vibrational STEM-EELS combined with DFT.Using the atomic displacement selectivity of off-axis EELS we observe that while the atomic displacements along z are different at the two interfaces, the atomic displacements along x are similar at both interfaces, which demonstrates vibrational anisotropy unique to each interface.DFT then unravels the characteristics of these modes, some of which are highly localized to the interface with large atomic displacements along z while others are delocalized with in-plane displacements in the layers and z displacements at the interface.These unique vibrations are a result of the strong coupling between charge, structure, and phonons in these and similar materials.We also note that other materials which lack centrosymmetry (for instance chiral materials) would also result in non-equivalent interfaces in a superlattice, making this experimental framework relevant to many novel materials systems.The nonequivalence of the interface vibrations parallels the decades of research on emergent electronic behavior and has implications on the thermal and optoelectronic tunability of materials.and 40 Torr, followed by AlN and (Al0.65,Ga0.35)Ndeposition with growth rates ~ 1.5 Å/s and ~1.6 Å/s, respectively.To obtain abrupt interfaces, a 6 second sweep-out time was introduced after each individual layer deposition.A total of 265 periods of AlN-(Al0.65Ga0.35)Nwas grown for reproducibility, wherein each individual period is ~10 nm thick, comprising of 15 Å AlN (~3 unit cells) and 85 Å of AlN-(Al0.65Ga0.35)N(~17 unit cells) per period.
28 Å and c = 20.30Å. Structural optimizations for the superlattice were carried out with an energy cutoff of 800 eV and a 9x9x2 Monkhorst-Pack k-point grid, continuing until forces on atoms were below 0.01 eV/Å.For the constituent bulk materials, AlN optimizations used an 800 eV cutoff with an 8x8x8 k-point grid, resulting in lattice constants a = 3.11 Å and c = 4.90 Å, while GaN optimizations employed a 800 eV cutoff with a denser 12x12x12 k-point grid, with resulting lattice constants of a = 3.19 Å and c = 5.19 Å.Both bulk optimizations converged to the same force criterion as the superlattice.

Figure 1 . 4 Figure 2 . 8 Figure 3 . 2 Figure S2 . 2 Figure S3 . 3 Figure S4 . 4 Figure S5 . 4 Figure S6 . 5 Figure S7 . 5 Figure S8 . 6 Figure S9 . 7 Figure 7 Figure S11 . 8 Figure S12 . 8 Figure S13 .
Figure 1.The non-equivalent structure of III-V heterostructure interfaces.(a) A ball-andstick model illustrating a single period of a (AlN)2-(GaN)2 superlattice.The superlattice period is indicated with the solid lines while dashed red lines mark the interfaces.(b) Structural units present in a single period of a superlattice, each of which are separated by grey lines and labeled on the right-hand side.(c) Bond length of the III-N dipole emphasizing the unique state of each structural unit.Red-dashed lines indicate the interface planes in (a), and black-dashed lines divide structural unit in (b).The average bond length for the AlN (blue) and GaN (green) layers provide a reference for the interface displacements.(d) A high-angle annular darkfield (HAADF) image of an AlN-(Al0.65Ga0.35)Nsuperlattice showing bright (Al0.65Ga0.35)Nlayers and dark AlN layers.The spatial coordinates referencing vectors normal to the SL (z) and parallel to the SL (x) are annotated.(e) Higher resolution HAADF image that focuses on a single period of the superlattice and emphasizes the interfaces.The top and bottom-interfaces of the (Al0.65Ga0.35)Nare labeled and annotated in cyan.(f) Schematic diagram of a STEM off-axis EELS experiment where the scattered probe is tilted away from the optic-axis (green dotted) to an off-axis condition such that the spectrometer entrance aperture collects scattering space qn, where n refers to x-and z-directions.(g,h) Off-axis STEM-EELS line profile.(g) ADF profile to provide context to where in the sample the off-axis spectra in (h) are being acquired from.Higher intensity regions in the ADF are (Al0.65Ga0.35)Nand lower intensity are AlN.Cyan annotations in (h) mark continuous intensity across two periods for the III and N bands.(i) Profiles integrated over the red and blue energy windows annotated in (h) with the extrema annotated with arrows and the center of the (Al0.65Ga0.35)with doted lines....... 4 Figure 2. Angularly and spatially resolved vibrational response of a polar heterostructure.Reference (a) PACBED pattern annotating two darkfield conditions for separate vibrational EELS spectrum images.Scattering conditions annotated with qz are normal to the interfaces and along the III-V dipole while qx are in-plane.Representative period-projected (b) ADF profile to provide context to where in the sample the spectra are being acquired, with higher intensity (I) being (Al0.65Ga0.35)Nand lower intensity AlN.Period-projected off-axis EELS line scans for the (c) qx and (d) qz scattering.The original scans are found in Figure S4.The EELS spectra intensities are normalized by multiplying by energy-loss (E) to remove the elastic background and power law scaled to bring out lower intensity features.Dipole PPhDOS along the (e) qx and (f) qz axes that has been spectrally blurred to the finite energy resolution of experiments.(g-j) Intensity profiles as a function of position across the projected period for specific energy slices.Intensities are normalized relative to AlN and scaled from ±1, i.e., zero being the intensity of vibrational response in the AlN layer.Grey dotted lines indicate the approximate position of the interfaces as determined by the full-width-half-max of the ADF signal (black dash-dot).................................. 8 Figure 3. Unraveling vibrational dynamics at non-equivalent interfaces using DFT.(a) Phonon dispersion with the color indicating the PR.(b) Layer PPR with the colors indicating the layer's relative contribution to the total PR.Red (R) indicates localization within the layers, green (T) at the top-interrace, and blue (b) at the bottom-interface.(d-g) Shows the displacements on a model (left) and the xz-projected displacement magnitudes for the modes circled in (a-c).The models and profiles are centered on the AlN instead of the GaN to emphasize the amplitude profile in the AlN.The interfaces at the top and bottom of the GaN are annotated with black dotted lines........................................................................................................................................................ 11